Integral polynomials with small discriminants and resultants

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• 作者：Victor Beresnevich^a ; ^{victor.beresnevich@york.ac.uk" class="auth_mail" title="E-mail the corresponding author} ; ^{vb8@york.ac.uk" class="auth_mail" title="E-mail the corresponding author} ; Vasili Bernik^b ; ^{bernik@im.bas-net.by" class="auth_mail" title="E-mail the corresponding author} ; ^{bernik.vasili@mail.ru" class="auth_mail" title="E-mail the corresponding author} ; Friedrich Gö ; tze^c ; ^{goetze@math.uni-bielefeld.de" class="auth_mail" title="E-mail the corresponding author}
• 关键词：11J83 ; 11J13 ; 11K60 ; 11K55
• 刊名：Advances in Mathematics
• 出版年：2016
• 出版时间：6 August 2016
• 年：2016
• 卷：298
• 期：Complete
• 页码：393-412
• 全文大小：452 K

文摘

Let n∈N$n\in \mathbb{N}$ be fixed, Q>1$Q>1$ be a real parameter and P_n(Q)${\mathcal{P}}_n(Q)$ denote the set of polynomials over Z$\mathbb{Z}$ of degree n and height at most Q  . In this paper we investigate the following counting problems regarding polynomials with small discriminant D(P)$D(P)$ and pairs of polynomials with small resultant R(P₁,P₂)$R(P_1,P_2)$:

(i)

given  0≤v≤n−1$0\le v\le n-1$and a sufficiently large Q, estimate the number of polynomials  P∈P_n(Q)$P\in {\mathcal{P}}_n(Q)$such that

$0<|D(P)|\le Q^{2n-2-2v};$

(ii)

given  0≤w≤n$0\le w\le n$and a sufficiently large Q, estimate the number of pairs of polynomials  P₁,P₂∈P_n(Q)$P_1,P_2\in {\mathcal{P}}_n(Q)$such that

0<|R(P₁,P₂)|≤Q^2n−2w.$0<|R(P_1,P_2)|\le Q^{2n-2w}.$

Our main results provide lower bounds within the context of the above problems. We believe that these bounds are best possible as they correspond to the solutions of naturally arising linear optimisation problems. Using a counting result for the number of rational points near planar curves due to R. C. Vaughan and S. Velani we also obtain the complementary optimal upper bound regarding the discriminants of quadratic polynomials.